Problem 79 - Chemistry The Science in Context

Question

What are the possible values of quantum number \(\ell\) when \(n=4 ?\)

Step-by-Step Solution

Answer

Answer: The possible values of the quantum number \(\ell\) when \(n = 4\) are \(0\), \(1\), \(2\), and \(3\).

Step 1: Understanding the quantum numbers

The principal quantum number (\(n\)) describes the energy level of an electron in an atom. The azimuthal quantum number (\(\ell\)) is related to the shape of the electron's orbital and ranges from \(0\) to \(n - 1\) with integer values.

Step 2: Calculating the possible values of \(\ell\) for \(n=4\)

Given that \(n = 4\), we can find the possible values of \(\ell\) using the range \(0\) to \(n - 1\). In this case, \(\ell\) can have integer values between \(0\) and \(4 - 1 = 3\).

Step 3: Listing the possible values of \(\ell\)

The possible values of the quantum number \(\ell\) when \(n = 4\) are \(0\), \(1\), \(2\), and \(3\).

Key Concepts

Principal Quantum Number

The principal quantum number, denoted as \(n\), signifies the energy level of an electron within an atom and is one of the fundamental quantum numbers in quantum mechanics. It mainly determines the overall energy of an electron and is always a positive integer. For example, if \(n = 1\), the electron is in the first main energy level closest to the nucleus. When \(n = 2\), it's in the second energy level, and so on.

• Determines the size and energy of the atomic orbital
• The larger the value of \(n\), the higher the energy level and the larger the orbital size
• Electrons with higher \(n\) are farther from the nucleus, meaning they have higher energy

Where \(n\) gets larger, the electrons it describes become more energetic and are farther from the atomic nucleus. The total number of electrons that can occupy an energy level is given by the formula \(2n^2\). This provides the basic structure for the electron configuration of an atom by filling from lower to higher energy levels.

Azimuthal Quantum Number

The azimuthal quantum number, represented by \(\ell\), is another important quantum number that defines the shape of an electron's orbital. It provides insight into the subshell where the electron can be found and is essential for understanding the electron's angular momentum. For a given principal quantum number \(n\), \(\ell\) can take on any integer value from \(0\) to \(n-1\).

• \(\ell = 0\) corresponds to an s orbital
• \(\ell = 1\) corresponds to a p orbital
• \(\ell = 2\) corresponds to a d orbital
• \(\ell = 3\) corresponds to an f orbital

Each value of \(\ell\) thus corresponds to a specific type of orbital which impacts the shape and orientation. For example, in the exercise given where \(n = 4\), \(\ell\) can be \(0\), \(1\), \(2\), or \(3\). This results in possible subshells being s, p, d, and f subshells, adding depth to the electron's configuration within an atom.

Electron Energy Levels

Electron energy levels define how electrons are structured around an atomic nucleus and are instrumental in the arrangement of electrons in atoms. Electrons are not randomly distributed; they occupy specific energy levels, each corresponding to a different principal quantum number \(n\).

• Energy levels are often visualized as the "shells" or "orbits" surrounding the nucleus
• Each energy level can hold a specific number of electrons, given by the formula \(2n^2\)
• Lower energy levels are filled before electrons are placed in higher levels, following the Aufbau principle

These energy levels are critical as they define the atom's chemical properties and interactions. The possible positions of an electron are dictated by both the principal quantum number \(n\) and the azimuthal quantum number \(\ell\), significantly influencing chemical behavior and bonding by determining the subshell in which an electron resides. Understanding these concepts provides a solid foundation for predicting how atoms will interact in chemical processes.